Recently, I saw some discussions on a forum, where a very responsible person took the animal to the vet after the young iguana broke its tail. It even turned out to be an open fracture.
It was amputated and all went well.
That's not always the case as I can say from own experience. In April this year, I saw that the tip of Zorro's tail was dying off. It seemed to be due to a hissy tail whip against some hard furniture and it needed amputation. A week after that, he whipped again and bruised the area at the remaining tip, so it died off further and needed another GA and amputation.
All went well, but it wouldn't heal. He was so aggressive that, every time he saw me, he whipped. In addition to that, the timing wasn't really good since he shed a few weeks later. He bled and the stitches came off, leaving the flesh and bone exposed.
I'm quite experienced, so I didn't panic, just cleaned it thoroughly and put some sterile bandage on. It wouldn't heal properly and took ages, because due to his whipping, he managed to get the bandage off quite a few times. I changed it every three days and finally, over night, left it open and it was sealed the next day.
It's all tip top now, the vet had a look at my recent visit and he was happy with it.
First aid tip for a broken tail in a younger iguana (or even toes):
This is only first aid to prevent the tail from further damage.
You need a straw (cut into pieces to go well over the break), tissue and desinfection.
Cut the straw-pieces so that you can bend them open. Cover the animal with a towel, if necessary, put it between your legs so it can't move.
Put the tissue carefully around the break, disinfect beforehand. Bend one straw lengthwise around the tissue and do the same with the other one. Fix it with plaster into place.
Make sure there's enough tissue under the straw and so that the skin is not damaged.
I personally don't see the vet immediately when I see it can be fixed by me. I managed to safe the tail of my female which was broken and only hung on one little string. It took ages, but it grew back together, healed and was not to see anymore after that.
If you are not experienced or unsure, always seek a vet's advice.